2010-2017 GM part # 13578630 - Driver Seat Horizontal Adjuster Actuator
At least these parts aren't expensive (I assume this is the one you might need).
As for that broken part ... it's interesting that it appears to have ball bearings embedded in it ... like it would be used to roll back and forth in the track ... so maybe that is your real problem?
Then again ... it doesn't look very substantial ... like it could take the beating it would need to take day in and day out.
I think you're gonna have to just remove the Seat to see what really happened underneath ... and maybe bench-test the motor to make sure it's not just binding up somehow ...